Is it possible to split a vertex?
Yes, you can just hit Alt + v to split it while hovering over the edge you would slide it on and then click g to slide it:



It's also in Mesh -> Vertices menu(Ctrl+v) as Rip Fill



Edit: I just realized this is not completely the desired result you show in the picture, because there is still an edge left where it was originally and you would need to dissolve it with Alt+x, however I think this is the closest thing. I think it would not make sense to have a tool that would remove the edge, because in that case it would in most cases leave untidy geometry, that might be best to avoid, so I can understand why there is no such a tool precisely as you described.